(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Biodiversity Heritage Library | Children's Library | Advanced Microdevices Manuals | Linear Circuits Manuals | Supertex Manuals | Sundry Manuals | Echelon Manuals | RCA Manuals | National Semiconductor Manuals | Hewlett Packard Manuals | Signetics Manuals | Fluke Manuals | Datel Manuals | Intersil Manuals | Zilog Manuals | Maxim Manuals | Dallas Semiconductor Manuals | Temperature Manuals | SGS Manuals | Quantum Electronics Manuals | STDBus Manuals | Texas Instruments Manuals | IBM Microsoft Manuals | Grammar Analysis | Harris Manuals | Arrow Manuals | Monolithic Memories Manuals | Intel Manuals | Fault Tolerance Manuals | Johns Hopkins University Commencement | PHOIBLE Online | International Rectifier Manuals | Rectifiers scrs Triacs Manuals | Standard Microsystems Manuals | Additional Collections | Control PID Fuzzy Logic Manuals | Densitron Manuals | Philips Manuals | The Andhra Pradesh Legislative Assembly Debates | Linear Technologies Manuals | Cermetek Manuals | Miscellaneous Manuals | Hitachi Manuals | The Video Box | Communication Manuals | Scenix Manuals | Motorola Manuals | Agilent Manuals
Search: Advanced Search
Anonymous User (login or join us)
See other formats

Full text of "Treatise On Analysis Vol-Ii"


the existence, write \JL = / p and v = g - p, where p is a positive measure and
/^ 0, and let M be the p-measurable set of points x e X such thatf(x) > 0, so
that \JL is concentrated on M (13.14.1). Then the measures v' = <pMg - p and
v" = v  v' = (1  <PM)# ' P satisfy the conditions of the theorem. For v" is
disjoint from \JL by virtue of (13.18.1); also if A is a /z-negligible set, then
A n M is p-negligible ((13.14.1) and (13.6,3)), hence A is v'-negligible
(13.14.1) and consequently v' is a measure with base \JL (13.15.5). It is clear that
if v is positive then so are v' and v" ; the formula v' = sup(inf(v, /7/z)) then


follows from the corresponding formula (pMg = sup(inf(#, /)), which is


valid almost everywhere with respect to p.

A measure /^ on X is said to be diffuse if n({x}) = 0 for all x E X. For
example, Lebesgue measure on R is diffuse. With respect to a diffuse measure
\JL every denumerable set is ^-negligible*, equivalently, a diffuse measure is
concentrated on the complement of any denumerable set. On a denumer-
able discrete space, the only diffuse measure is therefore n = 0. The sum of
two diffuse measures is diffuse (13.16.1). The least upper bound of any set
of diffuse positive measures which is bounded above is diffuse, by (13.15.9)
and (13. 15.4).

(13.18.5)   If \JL is any measure on X, the set A of points xeX such that
\I*\({x}) > 0 -is at most denumerable.

Since X is the union of a sequence of compact sets Kn , it is enough to show
that each of the sets A n Kn is at most denumerable. For this, it is enough to
show that, for each integer m JS> 1 , the set Amn of points x e A n Kn such that
M(M) ^ \l m infinite', now this is immediate, for if B c Amn consists of p
points, then we have p/m \p\(B) ^

A measure /z on X is said to be atomic if it is concentrated on an at most
denumerable set. From this definition and from (1 3.1 8.1 ) it follows immediately
that an atomic measure and a diffuse measure are always disjoint. The sum of
two atomic measures is atomic. The least upper bound of a set of atomic
positive measures which is bounded above is atomic ((13.15.4) and (13.15.9)).

(13.18.6)    Every measure ^ can be expressed uniquely in the form ^ + /x2,
where /^ is a diffuse measure and ju2 an atomic measure.

The uniqueness of the decomposition follows from the fact that an atomic
measure and a diffuse measure are disjoint. To establish the existence of
the decomposition, it is enough to consider the at most denumerable set fr)l fr ^1  e fri* ^i], then we have